let L be ExtREAL_sequence; :: thesis: for K being R_eal st K <> +infty & ( for n being Nat holds L . n <= K ) holds
sup (rng L) < +infty
let K be R_eal; :: thesis: ( K <> +infty & ( for n being Nat holds L . n <= K ) implies sup (rng L) < +infty )
assume that
A1: K <> +infty and
A2: for n being Nat holds L . n <= K ; :: thesis: sup (rng L) < +infty

now

let x be ext-real number ; :: thesis: ( x in rng L implies x <= K )
assume x in rng L ; :: thesis: x <= K
then ex z being set st
( z in dom L & x = L . z ) by FUNCT_1:def 5;
hence x <= K by A2; :: thesis: verum

end;

then K is UpperBound of rng L by XXREAL_2:def 1;
then sup (rng L) <= K by XXREAL_2:def 3;
hence sup (rng L) < +infty by A1, XXREAL_0:2, XXREAL_0:4; :: thesis: verum